StudyBrixFree worksheets, study guides and courses

Mathematics · lesson 3 of 4

Gradients, tangent planes and extrema

The gradient collects first partial derivatives and determines directional derivatives through a dot product.

University8 practice questionsNotes + worked examples
Take the quizStudy guideRevision sheet

Learn the ideas

Prerequisites: Partial derivatives and vector dot products.

Gradient

The gradient collects first partial derivatives and determines directional derivatives through a dot product.

Tangent plane

Linearization at a differentiable point uses f(a,b)+fₓ(a,b)(x−a)+fᵧ(a,b)(y−b).

Stationary points

A zero gradient is necessary for an interior differentiable extremum, not sufficient.

Hessian test

In two variables, positive determinant and positive fₓₓ identify a strict local minimum; negative determinant identifies a saddle.

Worked examples

Example 1

For f(x,y)=x²+3y², find fₓ at (2,1).

Differentiate with respect to x to obtain 2x, then substitute x=2.

4

Example 2

Write the tangent plane to z=x²+y² at (1,2,5).

Credit the correct base height and both gradient coefficients.

z=5+2(x−1)+4(y−2), equivalently z=2x+4y−5.

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 For f(x,y)=x²+3y², find fₓ at (2,1).

Show answer and explanation

4

Differentiate with respect to x to obtain 2x, then substitute x=2.

2 Write the tangent plane to z=x²+y² at (1,2,5).

Sample answer and marking guidance

z=5+2(x−1)+4(y−2), equivalently z=2x+4y−5.

Credit the correct base height and both gradient coefficients.

3 For f(x,y)=x²+3y², find the directional derivative at (1,1) in direction (3/5,4/5).

Show answer and explanation

6

The gradient is (2,6). Its dot product with (3/5,4/5) is 6/5+24/5=6.

4 What kind of stationary point is the origin for f=x²−y²?

Show answer and explanation

A saddle point

The function increases along the x-axis and decreases along the y-axis.

5 Find the determinant of the Hessian of f=2x²+3xy+5y².

Show answer and explanation

31

The Hessian is [[4,3],[3,10]]; its determinant is 40−9=31.

6 Why should the direction vector be normalized before calculating a directional derivative?

Sample answer and marking guidance

A directional derivative is conventionally the rate per unit distance in that direction. Without normalization, the dot product is scaled by the vector length.

Credit the unit-distance interpretation and the scaling effect.

7 If a stationary point has Hessian determinant zero, what does the ordinary second-derivative test conclude?

Show answer and explanation

It is inconclusive

Higher-order terms or another argument may be needed.

8 Give a differentiable function whose gradient is zero at the origin but that has no local extremum there.

Sample answer and marking guidance

For example f(x,y)=x²−y² has gradient zero at the origin but takes both positive and negative values arbitrarily close to it.

Credit a valid example and evidence that neither minimum nor maximum holds.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.